There are two natural ideas that appear in Statistics, the standard deviation and the maximum entropy principle. If you ask the question, “Among all distributions with standard deviation 1 and mean 0, what is the distribution with maximum entropy?” The answer is the Gaussian.

Randomly select a point inside a high dimensional hypersphere. The distribution of any particular coordinate is approximately Gaussian. The same is true for a random point on the surface of the hypersphere.

Take several samples from a Gaussian Distribution. Compute the Discrete Fourier Transform of the samples. The results have a Gaussian Distribution. I am pretty sure that the Gaussian is the only distribution with this property.

“The normal distribution is the only absolutely continuous distribution all of whose cumulants beyond the first two (i.e. other than the mean and variance) are zero.” – Wikipedia.

For even n, the nth moment of the Gaussian is simply an integer multiplied by the standard deviation to the nth power.

Many of the other standard distributions are strongly related to the Gaussian (i.e. binomial, Poisson, chi-squared, Student t, Rayleigh, Logistic, Log-Normal, Hypergeometric …)

“If X1 and X2 are independent and their sum X1 + X2 is distributed normally, then both X1 and X2 must also be normal.” — From the Wikipedia.

“The conjugate prior of the mean of a normal distribution is another normal distribution.” — From the Wikipedia.

When using Gaussians, the math is easier.

The Erdős–Kac theorem implies that the distribution of the prime factors of a “random” integer is Gaussian.

The velocities of random molecules in a gas are distributed as a Gaussian. (With standard deviation = $z*\sqrt{ k\, T / m} $ where $z$ is a “nice” constant, $m$ is the mass of the particle, and $k$ is Boltzmann’s constant.)

“A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator.” — From Wikipedia